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Leaf as a Poincaré convex domain associated with an endomorphism on a real inner product space

Hiroyuki Ogawa

TL;DR

The paper introduces the leaf Ψ(phi), a geometric object defined via a Rayleigh-type map on a real inner product space, as a subset of the closure of the upper half-plane equipped with the Poincaré metric. It proves that for dim V ≥ 3 the leaf is convex in the Poincaré metric and contains all eigenvalues with nonnegative imaginary part, with the smallest such convex domain given by the eigenvalue geodesic polygon; in the normal case, the leaf coincides with this polygon. The work contrasts the leaf with the classical numerical range, highlighting that the leaf preserves information about real eigenvalues and uses hyperbolic geometry to visualize endomorphisms, including explicit low-dimensional classifications (dimension 1 or 2) and a general framework for higher dimensions. It further shows that leaf structure is robust under decomposition into φ-stable subspaces and that, for normal operators, the leaf aligns with the spectral geometry of eigenvalues, while non-normal operators can yield larger leaves sharing the same geometry. Overall, the paper provides a new hyperbolic-geometric viewpoint on endomorphisms and spectral structure, offering a visual and structural bridge between eigenvalues and convex geometric concepts.

Abstract

We define a subset of the closure of the upper half plane associated with an endomorphism on a real inner product space, which is called the leaf. When the dimension of the space is at least 3, the leaf is a convex with respect to the Poincaré metric, and contains all eigenvalues with nonnegative imaginary part. Moreover, the leaf of a normal endomorphism is the minimum Poincaré convex domain containing all eigenvalues with nonnegative imaginary part. The most commonly studied convex domain containing eigenvalues is number range. Numerical range is convex with respect to the Euclidean metric on $\mathbb C$, so numerical range has less information than leaf about real eigenvalues. We provide a new visual approach to endomorphisms.

Leaf as a Poincaré convex domain associated with an endomorphism on a real inner product space

TL;DR

The paper introduces the leaf Ψ(phi), a geometric object defined via a Rayleigh-type map on a real inner product space, as a subset of the closure of the upper half-plane equipped with the Poincaré metric. It proves that for dim V ≥ 3 the leaf is convex in the Poincaré metric and contains all eigenvalues with nonnegative imaginary part, with the smallest such convex domain given by the eigenvalue geodesic polygon; in the normal case, the leaf coincides with this polygon. The work contrasts the leaf with the classical numerical range, highlighting that the leaf preserves information about real eigenvalues and uses hyperbolic geometry to visualize endomorphisms, including explicit low-dimensional classifications (dimension 1 or 2) and a general framework for higher dimensions. It further shows that leaf structure is robust under decomposition into φ-stable subspaces and that, for normal operators, the leaf aligns with the spectral geometry of eigenvalues, while non-normal operators can yield larger leaves sharing the same geometry. Overall, the paper provides a new hyperbolic-geometric viewpoint on endomorphisms and spectral structure, offering a visual and structural bridge between eigenvalues and convex geometric concepts.

Abstract

We define a subset of the closure of the upper half plane associated with an endomorphism on a real inner product space, which is called the leaf. When the dimension of the space is at least 3, the leaf is a convex with respect to the Poincaré metric, and contains all eigenvalues with nonnegative imaginary part. Moreover, the leaf of a normal endomorphism is the minimum Poincaré convex domain containing all eigenvalues with nonnegative imaginary part. The most commonly studied convex domain containing eigenvalues is number range. Numerical range is convex with respect to the Euclidean metric on , so numerical range has less information than leaf about real eigenvalues. We provide a new visual approach to endomorphisms.
Paper Structure (8 sections, 24 theorems, 53 equations, 1 figure)

This paper contains 8 sections, 24 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Normal endomorphisms
  3. The Poincaré metric on ${\mathfrak{H}}$
  4. Leaf and real eigenvalues
  5. Numerical range vs. leaf
  6. Leaf of an endomorphism of dimension $1$ or $2$
  7. Leaf in $\overline{{\mathfrak{H}}}$ and eigenvalues with nonnegative imaginary part
  8. Leaf and the eigenvalue geodesic polygon

Key Result

Theorem A

Leaf is convex with respect to the Poincaré metric.

Figures (1)

  • Figure 1: Examples of leaves

Theorems & Definitions (45)

  • Definition 1.1
  • Example 1.2
  • Theorem A: Poincaré Convexity
  • Theorem B: Eigenvalue inclusion
  • Theorem C: Normal case
  • Remark 1.3
  • Definition 4.1: leaf
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • ...and 35 more